Null quantization, shadows and boost eigenfunctions in Lorentzian AdS

We revisit the quantization of a free scalar in 4-dimensional (4d) Lorentzian Anti-de-Sitter spacetime (AdS$_4$). We derive solutions to the wave equation that diagonalize time translations in a foliation of AdS$_4$ with null cones. We show that time-translation eigenmodes of arbitrary mass fields that admit a flat space limit must contain both normalizable and non-normalizable fall-offs as one approaches the boundary along a null leaf. We then show that AdS bulk-to-boundary propagators with suitable time orderings provide alternative bases of solutions to the wave equation. We propose an AdS bulk reconstruction formula relating an on-shell free scalar at a spacetime point to CFT primary operators and their shadow transforms. In the flat space limit, this formula reduces to the Carrollian expansion of a free field in flat space. We finally construct Lorentz boost eigenfunctions in AdS in both hyperbolic and null foliations and show that they respectively become massive and massless conformal primary wavefunctions in the flat space limit.